# Probability Density Function of the Reverse Gumbel Distribution

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### Description

This function computes the probability density of the Reverse Gumbel distribution given parameters (ξ and α) computed by parrevgum. The probability density function is

f(x) = α^{-1} \exp(Y) [\exp(\exp[-\exp(Y)])] \mbox{,}

where

Y = \frac{x - ξ}{α} \mbox{,}

where f(x) is the probability density for quantile x, ξ is a location parameter, and α is a scale parameter.

### Usage

 1 pdfrevgum(x, para) 

### Arguments

 x A real value vector. para The parameters from parrevgum or vec2par.

### Value

Probability density (f) for x.

W.H. Asquith

### References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

cdfrevgum, quarevgum, lmomrevgum, parrevgum
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 # See p. 553 of Hosking (1995) # Data listed in Hosking (1995, table 29.3, p. 553) D <- c(-2.982, -2.849, -2.546, -2.350, -1.983, -1.492, -1.443, -1.394, -1.386, -1.269, -1.195, -1.174, -0.854, -0.620, -0.576, -0.548, -0.247, -0.195, -0.056, -0.013, 0.006, 0.033, 0.037, 0.046, 0.084, 0.221, 0.245, 0.296) D <- c(D,rep(.2960001,40-28)) # 28 values, but Hosking mentions # 40 values in total z <- pwmRC(D,threshold=.2960001) str(z) # Hosking reports B-type L-moments for this sample are # lamB1 = -0.516 and lamB2 = 0.523 btypelmoms <- pwm2lmom(z$Bbetas) # My version of R reports lamB1 = -0.5162 and lamB2 = 0.5218 str(btypelmoms) rg.pars <- parrevgum(btypelmoms,z$zeta) str(rg.pars) # Hosking reports xi=0.1636 and alpha=0.9252 for the sample # My version of R reports xi = 0.1635 and alpha = 0.9254 # Now one can continue one with a plotting example. ## Not run: F <- nonexceeds() PP <- pp(D) # plotting positions of the data D <- sort(D) plot(D,PP) lines(D,cdfrevgum(D,rg.pars)) # Now finally do the PDF F <- seq(0.01,0.99,by=.01) x <- quarevgum(F,rg.pars) plot(x,pdfrevgum(x,rg.pars),type='l') ## End(Not run)